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5.7 Graphing and Solving Quadratic Inequalities 299

Graphing and SolvingQuadratic Inequalities

QUADRATIC INEQUALITIES IN TWO VARIABLES

In this lesson you will study four types of

y < ax2 + bx + c y ≤ ax2 + bx + c

y > ax2 + bx + c y ≥ ax2 + bx + c

The graph of any such inequality consists of all solutions (x, y) of the inequality. The steps used to graph a quadratic inequality are very much like those used to grapha linear inequality. (See Lesson 2.6.)

Graphing a Quadratic Inequality

Graph y > x2 º 2x º 3.

SOLUTION

Follow Steps 1–3 listed above.

Graph y = x2 º 2x º 3. Since the inequalitysymbol is >, make the parabola dashed.

Test a point inside the parabola, such as (1, 0).

y > x2 º 2x º 3

0 >?

12 º 2(1) º 3

0 > º4 ✓

So, (1, 0) is a solution of the inequality.

Shade the region inside the parabola.3

2

1

E X A M P L E 1

quadratic inequalities in two variables.

GOAL 1

Graph quadraticinequalities in two variables.

Solve quadraticinequalities in one variable,as applied in Example 7.

� To solve real-lifeproblems, such as finding theweight of theater equipmentthat a rope can support in Exs. 47 and 48.

Why you should learn it

GOAL 2

GOAL 1

What you should learn

5.7RE

AL LIFE

RE

AL LIFE To graph one of the four types of quadratic inequalities shown above, followthese steps:

STEP Draw the parabola with equation y = ax2 + bx + c. Make the paraboladashed for inequalities with < or > and solid for inequalities with ≤ or ≥.

STEP Choose a point (x, y) inside the parabola and check whether the pointis a solution of the inequality.

STEP If the point from Step 2 is a solution, shade the region inside theparabola. If it is not a solution, shade the region outside the parabola.

3

2

1

GRAPHING A QUADRATIC INEQUALITY IN TWO VARIABLES

4

1

(1, 0)

y

x

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300 Chapter 5 Quadratic Functions

Using a Quadratic Inequality as a Model

You are building a wooden bookcase. You want tochoose a thickness d (in inches) for the shelves sothat each is strong enough to support 60 pounds ofbooks without breaking. A shelf can safely supporta weight of W (in pounds) provided that:

W ≤ 300d2

a. Graph the given inequality.

b. If you make each shelf 0.75 inch thick, can it support a weight of 60 pounds?

SOLUTION

a. Graph W = 300d2 for nonnegative values of d. Since the inequality symbol is ≤, make the parabola solid. Test a point inside the parabola, such as (0.5, 240).

W ≤ 300d2

240 ≤?

300(0.5)2

240 � 75

Since the chosen point is not a solution, shadethe region outside (below) the parabola.

b. The point (0.75, 60) lies in the shaded region of the graph from part (a), so (0.75, 60) is a solution of the given inequality. Therefore, a shelf that is 0.75 inchthick can support a weight of 60 pounds.

. . . . . . . . . .

Graphing a system of quadratic inequalities is similar to graphing a system of linearinequalities. First graph each inequality in the system. Then identify the region in thecoordinate plane common to all the graphs. This region is called the graph of the system.

Graphing a System of Quadratic Inequalities

Graph the system of quadratic inequalities.

y ≥ x2 º 4 Inequality 1

y < ºx2 º x + 2 Inequality 2

SOLUTION

Graph the inequality y ≥ x2 º 4. The graph is the redregion inside and including the parabola y = x2 º 4.

Graph the inequality y < ºx2 º x + 2. The graph is the blue region inside (but not including) the parabolay = ºx2 º x + 2.

Identify the purple region where the two graphs overlap.This region is the graph of the system.

E X A M P L E 3

E X A M P L E 2RE

AL LIFE

REAL LIFE

Carpentry

12 in.

48 in.

d in.

200250300 (0.5, 240)

W ≤ 300d 2

0

100150

0 0.5 1.0 1.5

Safe

wei

ght (

lb)

Thickness (in.)

(0.75, 60)

d

W

50

3

y

1

y < �x 2 � x � 2

y ≥ x 2 � 4

x

Look Back For help with graphinginequalities in twovariables, see p. 108.

STUDENT HELP

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5.7 Graphing and Solving Quadratic Inequalities 301

QUADRATIC INEQUALITIES IN ONE VARIABLE

One way to solve a is to use a graph.

• To solve ax2 + bx + c < 0 (or ax2 + bx + c ≤ 0), graph y = ax2 + bx + c andidentify the x-values for which the graph lies below (or on and below) the x-axis.

• To solve ax2 + bx + c > 0 (or ax2 + bx + c ≥ 0), graph y = ax2 + bx + c andidentify the x-values for which the graph lies above (or on and above) the x-axis.

Solving a Quadratic Inequality by Graphing

Solve x2 º 6x + 5 < 0.

SOLUTION

The solution consists of the x-values for which the graph of y = x2 º 6x + 5 lies below the x-axis. Find the graph’s x-intercepts by letting y = 0 and using factoring to solve for x.

0 = x2 º 6x + 5

0 = (x º 1)(x º 5)

x = 1 or x = 5

Sketch a parabola that opens up and has 1 and 5 as x-intercepts. The graph lies below the x-axis between x = 1 and x = 5.

� The solution of the given inequality is 1 < x < 5.

Solving a Quadratic Inequality by Graphing

Solve 2x2 + 3x º 3 ≥ 0.

SOLUTION

The solution consists of the x-values for which thegraph of y = 2x2 + 3x º 3 lies on and above the x-axis. Find the graph’s x-intercepts by letting y = 0and using the quadratic formula to solve for x.

0 = 2x2 + 3x º 3

x =

x = �º3 ±4

�3�3��

x ≈ 0.69 or x ≈ º2.19

Sketch a parabola that opens up and has 0.69 and º2.19 as x-intercepts. The graphlies on and above the x-axis to the left of (and including) x = º2.19 and to the rightof (and including) x = 0.69.

� The solution of the given inequality is approximately x ≤ º2.19 or x ≥ 0.69.

º3 ± �3�2�º� 4�(2�)(�º�3�)����

2(2)

E X A M P L E 5

E X A M P L E 4

quadratic inequality in one variable

GOAL 2

4

1

y

y � 2x 2 � 3x � 3

�2.19 0.69 x

y

1

y � x 2 � 6x � 5

51x3

Look Back For help with solvinginequalities in onevariable, see p. 41.

STUDENT HELP

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302 Chapter 5 Quadratic Functions

You can also use an algebraic approach to solve a quadratic inequality in onevariable, as demonstrated in Example 6.

Solving a Quadratic Inequality Algebraically

Solve x2 + 2x ≤ 8.

SOLUTION

First write and solve the equation obtained by replacing the inequality symbol withan equals sign.

x2 + 2x ≤ 8 Write original inequality.

x2 + 2x = 8 Write corresponding equation.

x2 + 2x º 8 = 0 Write in standard form.

(x + 4)(x º 2) = 0 Factor.

x = º4 or x = 2 Zero product property

The numbers º4 and 2 are called the critical x-values of the inequality x2 + 2x ≤ 8.Plot º4 and 2 on a number line, using solid dots because the values satisfy theinequality. The critical x-values partition the number line into three intervals. Test an x-value in each interval to see if it satisfies the inequality.

Test x = º5: Test x = 0: Test x = 3:(º5)2 + 2(º5) = 15 � 8 02 + 2(0) = 0 ≤ 8 ✓ 32 + 2(3) = 15 � 8

� The solution is º4 ≤ x ≤ 2.

Using a Quadratic Inequality as a Model

DRIVING For a driver aged x years, a study found that the driver’s reaction time V(x)(in milliseconds) to a visual stimulus such as a traffic light can be modeled by:

V(x) = 0.005x2 º 0.23x + 22, 16 ≤ x ≤ 70

At what ages does a driver’s reaction time tend to be greater than 25 milliseconds?� Source: Science Probe!

SOLUTION

You want to find the values of x for which:

V(x) > 25

0.005x2 º 0.23x + 22 > 25

0.005x2 º 0.23x º 3 > 0

Graph y = 0.005x2 º 0.23x º 3 on the domain 16 ≤ x ≤ 70. The graph’s x-interceptis about 57, and the graph lies above the x-axis when 57 < x ≤ 70.

� Drivers over 57 years old tend to have reaction times greater than 25 milliseconds.

E X A M P L E 7

�6 �5 �4 �3 �2 �1 0 1 2 3 4

E X A M P L E 6

ZeroX=56.600595 Y=0

DRIVING Drivingsimulators help

drivers safely improve theirreaction times to hazardoussituations they mayencounter on the road.

RE

AL LIFE

RE

AL LIFE

FOCUS ONAPPLICATIONS

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5.7 Graphing and Solving Quadratic Inequalities 303

1. Give one example each of a quadratic inequality in one variable and a quadraticinequality in two variables.

2. How does the graph of y > x2 differ from the graph of y ≥ x2?

3. Explain how to solve x2 º 3x º 4 > 0 graphically and algebraically.

Graph the inequality.

4. y ≥ x2 + 2 5. y ≤ º2x2 6. y < x2 º 5x + 4

Graph the system of inequalities.

7. y ≤ ºx2 + 3 8. y ≥ ºx2 + 3 9. y ≥ ºx2 + 3y ≥ x2 + 2x º 4 y ≥ x2 + 2x º 4 y ≤ x2 + 2x º 4

Solve the inequality.

10. x2 º 4 < 0 11. x2 º 4 ≥ 0 12. x2 º 4 > 3x

13. ARCHITECTURE The arch of the Sydney Harbor Bridge in Sydney,Australia, can be modeled by y = º0.00211x2 + 1.06x where x is the distance(in meters) from the left pylons and y is the height (in meters) of the arch abovethe water. For what distances x is the arch above the road?

MATCHING GRAPHS Match the inequality with its graph.

14. y ≥ x2 º 4x + 1 15. y < x2 º 4x + 1 16. y ≤ ºx2 º 4x + 1

A. B. C.

GRAPHING QUADRATIC INEQUALITIES Graph the inequality.

17. y ≥ 3x2 18. y ≤ ºx2 19. y > ºx2 + 5

20. y < x2 º 3x 21. y ≤ x2 + 8x + 16 22. y ≤ ºx2 + x + 6

23. y ≥ 2x2 º 2x º 5 24. y ≥ º2x2 º x + 3 25. y > º3x2 + 5x º 4

26. y < º�12�x2 º 2x + 4 27. y > �

43�x2 º 12x + 29 28. y < 0.6x2 + 3x + 2.4

y

1

1

x

1

1

y

x

y5

2 x

PRACTICE AND APPLICATIONS

GUIDED PRACTICE

Vocabulary Check ✓

Concept Check ✓

Skill Check ✓

Extra Practiceto help you masterskills is on p. 947.

STUDENT HELP

STUDENT HELP

HOMEWORK HELPExample 1: Exs. 14–28Example 2: Exs. 47–49Example 3: Exs. 29–34,

49Examples 4, 5: Exs. 35–40Example 6: Exs. 41–46Example 7: Exs. 50, 51

y

x

52 m

pylon

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304 Chapter 5 Quadratic Functions

GRAPHING SYSTEMS Graph the system of inequalities.

29. y ≥ x2 30. y < º3x2 31. y > x2 º 6x + 9y ≤ x2 + 3 y ≥ º�

12�x2 º 5 y < ºx2 + 6x º 3

32. y ≥ x2 + 2x + 1 33. y < 3x2 + 2x º 5 34. y ≤ 2x2 º 9x + 8y ≥ x2 º 4x + 4 y ≥ º2x2 + 1 y > ºx2 º 6x º 4

SOLVING BY GRAPHING Solve the inequality by graphing.

35. x2 + x º 2 < 0 36. 2x2 º 7x + 3 ≥ 0 37. ºx2 º 2x + 8 ≤ 0

38. ºx2 + x + 5 > 0 39. 3x2 + 24x ≥ º41 40. º�34�x2 + 4x º 8 < 0

SOLVING ALGEBRAICALLY Solve the inequality algebraically.

41. x2 + 3x º 18 ≥ 0 42. 3x2 º 16x + 5 ≤ 0 43. 4x2 < 25

44. ºx2 º 12x < 32 45. 2x2 º 4x º 5 > 0 46. �12�x2 + 3x ≤ º6

THEATER In Exercises 47 and 48, use the following information. You are a member of a theater production crew. You use manila rope and wire rope to support lighting, scaffolding, and other equipment. The weight W (in pounds) thatcan be safely supported by a rope with diameter d (in inches) is given below for bothtypes of rope. � Source: Workshop Math

Manila rope: W ≤ 1480d2 Wire rope: W ≤ 8000d2

47. Graph the inequalities in separate coordinate planes for 0 ≤ d ≤ 1�12�.

48. Based on your graphs, can 1000 pounds of theater equipment be supported by

a �12� inch manila rope? by a �

12� inch wire rope?

49. HEALTH For a person of height h (in inches), a healthy weight W(in pounds) is one that satisfies this system of inequalities:

W ≥ �1790h3

2� and W ≤ �27

50h3

2�

Graph the system for 0 ≤ h ≤ 80. What is the range of healthy weights for aperson 67 inches tall? � Source: Parade Magazine

SOLVING INEQUALITIES In Exercises 50–52, you may want to use agraphing calculator to help you solve the problems.

50. FORESTRY Sawtimber is a term for trees that are suitable for sawing intolumber, plywood, and other products. For the years 1983–1995, the unit value y(in 1994 dollars per million board feet) of one type of sawtimber harvested inCalifornia can be modeled by

y = 0.125x2 º 569x + 848,000, 400 ≤ x ≤ 2200

where x is the volume of timber harvested (in millions of board feet).� Source: California Department of Forestry and Fire Protection

a. For what harvested timber volumes is the value of the timber at least $400,000per million board feet?

b. LOGICAL REASONING What happens to the unit value of the timber as thevolume harvested increases? Why would you expect this to happen?

SET DESIGNERA set designer

creates the scenery, or sets,used in a theater production.The designer may makescale models of the setsbefore they are actuallybuilt.

CAREER LINKwww.mcdougallittell.com

INTE

RNET

RE

AL LIFE

RE

AL LIFE

FOCUS ONCAREERS

HOMEWORK HELPVisit our Web site

www.mcdougallittell.comfor help with problemsolving in Exs. 50–52.

INTE

RNET

STUDENT HELP

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5.7 Graphing and Solving Quadratic Inequalities 305

51. MEDICINE In 1992 the average income I (in dollars) for a doctor aged x years could be modeled by:

I = º425x2 + 42,500x º 761,000

For what ages did the average income for a doctor exceed $250,000?DATA UPDATE of American Almanac of Jobs and Salaries data at www.mcdougallittell.com

52. MULTI-STEP PROBLEM A study of driver reaction times to audio stimuli foundthat the reaction time A(x) (in milliseconds) of a driver can be modeled by

A(x) = 0.0051x2 º 0.319x + 15, 16 ≤ x ≤ 70

where x is the driver’s age (in years). � Source: Science Probe!

a. Graph y = A(x) on the given domain. Also graph y = V(x), the reaction-timemodel for visual stimuli from Example 7, in the same coordinate plane.

b. For what values of x in the interval 16 ≤ x ≤ 70 is A(x) < V(x)?

c. Writing Based on your results from part (b), do you think a driver wouldreact more quickly to a traffic light changing from green to yellow or to thesiren of an approaching ambulance? Explain.

53. The area A of the region bounded by a parabola and a horizontal line is given by

A = �23�bh where b and h are as defined in the diagram.

Find the area of the region determined by each pair of inequalities.

a. y ≤ ºx2 + 4x b. y ≥ x2 º 4x º 5y ≥ 0 y ≤ 3

SOLVING FOR A VARIABLE Solve the equation for y. (Review 1.4)

54. 3x + y = 1 55. 8x º 2y = 10 56. º2x + 5y = 9

57. �16�x + �

13�y = º�

11

12� 58. xy º x = 2 59. �

x º4

3y� = 7x

SOLVING SYSTEMS Solve the system of linear equations. (Review 3.6 for 5.8)

60. 5x º 3y º 2z = º17 61. x º 4y + z = º14ºx + 7y º 3z = 6 2x + 3y + 7z = º153x + 2y + 4z = 13 º3x + 5y º 5z = 29

COMPLEX NUMBERS Write the expression as a complex number in standardform. (Review 5.4)

62. (3 + 4i) + (10 º i) 63. (º11 º 2i) + (5 + 2i)

64. (9 + i) º (4 º i) 65. (5 º 3i) º (º1 + 2i)

66. 6i(8 + i) 67. (7 + 3i)(2 º 5i)

68. �3 º1

i� 69. �94

+º

23ii

�

MIXED REVIEW

GEOMETRY CONNECTION

INTE

RNET

TestPreparation

★★ Challenge

EXTRA CHALLENGE

www.mcdougallittell.com

x

y

b

h